Let $a_n $ complex sequence prove that if $ a_n\to \infty$ then $|a_n|\to\infty$. Note that $a_n = x_n + y_ni$

Question

Let $a_n $ complex sequence prove that if $ a_n\to \infty$ then $|a_n|\to\infty$. Note that $a_n = x_n + y_ni$ i dont know how to write that mathmatically.

trial :

Can i say that for every $M>0$ there exist $N$ such that for every $n>N$ ,

$~~|x_n|>M~~ OR ~~~|y_n|>M$ ( At least one of them goes to $\infty$)

because of that $|an| = \sqrt{(x_n)^2+(y_n)^2} > M$ and so $|a_n|\to\infty$.